Regularity of solutions of convolution equations

This paper investigates the regularity of solutions of convolution equations in the frame of classes of ultradifferentiable functions and ultradistributions. We improve previous work by Bonet, Chou, Fernández, Galbis, Meise and others.     

Density estimates for a random noise propagating through a chain of differential equations

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the n−1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on … , so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.     

Failure of hypoellipticity for the canonical solution to $${\bar\partial_b}$$ on some nilpotent Lie groups

In this paper we show that Kohn’s solution to the \({\bar\partial_b}\) problem fails to be hypoelliptic on some class of high codimension submanifolds of \({\mathbb{C}^n}\). The examples presented here carry a Lie group structure which generalize the one-dimensional Heisenberg group.     

Probabilistic solution of the American options

The existence and uniqueness of probabilistic solutions of variational inequalities for the general American options are proved under the hypothesis of hypoellipticity of the infinitesimal generator of the underlying diffusion process which represents the risky assets of the stock market with which the option is created. The main tool is an extension of the Itô formula which is valid for the tempered distributions on Rd and for nondegenerate Itô processes in the sense of the Malliavin calculus.     

Analytic hypoellipticity in the presence of nonsymplectic characteristic points

Recently, N. Hanges proved that the operator

     

Analyticity of solutions of semi-linear equations with double characteristics

We prove the analyticity and Gevrey regularity of solutions of elliptic degenerate semi-linear differential equations principle part of which is a linear operator with double characteristics considered first by Gilioli and Treves. A new elementary proof for hypoellipticity in the weak sense is given.     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation

     

A Gentle Stochastic Thermostat for Molecular Dynamics

We discuss a dynamical technique for sampling the canonical measure in molecular dynamics. We present a method that generalizes a recently proposed scheme (Samoletov et al., J. Stat. Phys. 128:1321–1336, 2007), and which controls temperature by use of a device similar to that of Nosé dynamics, but adds random noise to improve ergodicity. In contrast to Langevin dynamics, where noise is added directly to each physical degree of freedom, the new scheme relies on an indirect coupling to a single Brownian particle. For a model with harmonic potentials, we show under a mild non-resonance assumption that we can recover the canonical distribution. In spite of its stochastic nature, experiments suggest that it introduces a relatively weak perturbative effect on the physical dynamics, as measured by perturbation of temporal autocorrelation functions. The kinetic energy is well controlled even in the early stages of a simulation.